## Precalculus (6th Edition) Blitzer

The $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$ is $\frac{8}{5}\mathbf{i}+\frac{4}{5}\mathbf{j}$ and ${{\mathbf{v}}_{\mathbf{1}}}=\frac{8}{5}\mathbf{i}+\frac{4}{5}\mathbf{j}$, ${{\mathbf{v}}_{\mathbf{2}}}=\frac{7}{5}\mathbf{i}-\frac{14}{5}\mathbf{j}$.
The projection vector, $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$ can be obtained as, \begin{align} & \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v=}\frac{\mathbf{v}\cdot \mathbf{w}}{{{\left| \mathbf{w} \right|}^{2}}}\mathbf{w} \\ & =\frac{\left( 3\mathbf{i}-2\mathbf{j} \right)\cdot \left( 2\mathbf{i}+\mathbf{j} \right)}{{{\left( \sqrt{{{2}^{2}}+{{1}^{2}}} \right)}^{2}}}\left( 2\mathbf{i}+\mathbf{j} \right) \\ & =\frac{3\cdot 2+\left( -2 \right)\cdot 1}{5}\left( 2\mathbf{i}+\mathbf{j} \right) \\ & =\frac{6-2}{5}\left( 2\mathbf{i}+\mathbf{j} \right) \end{align} Solve ahead to get the result as, \begin{align} & \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{6-2}{5}\left( 2\mathbf{i}+\mathbf{j} \right) \\ & =\frac{4}{5}\left( 2\mathbf{i}+\mathbf{j} \right) \\ & =\frac{8}{5}\mathbf{i}+\frac{4}{5}\mathbf{j} \end{align} Now, obtain ${{\mathbf{v}}_{\mathbf{1}}}$ such that ${{\mathbf{v}}_{\mathbf{1}}}$ is parallel to $\mathbf{w}$ as, \begin{align} & {{\mathbf{v}}_{\mathbf{1}}}\mathbf{=}\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v} \\ & =\frac{8}{5}\mathbf{i}+\frac{4}{5}\mathbf{j} \end{align} Obtain ${{\mathbf{v}}_{\mathbf{2}}}$ such that ${{\mathbf{v}}_{\mathbf{2}}}$ is orthogonal to $\mathbf{w}$ as, \begin{align} & {{\mathbf{v}}_{\mathbf{2}}}=\mathbf{v}-{{\mathbf{v}}_{\mathbf{1}}} \\ & =\left( 3\mathbf{i}-2\mathbf{j} \right)-\left( \frac{8}{5}\mathbf{i}+\frac{4}{5}\mathbf{j} \right) \\ & =3\mathbf{i}-2\mathbf{j}-\frac{8}{5}\mathbf{i}-\frac{4}{5}\mathbf{j} \\ & =\frac{7}{5}\mathbf{i}-\frac{14}{5}\mathbf{j} \end{align} Hence, the projection vector, $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$ is $\frac{5}{2}\mathbf{i}-\frac{5}{2}\mathbf{j}$ and ${{\mathbf{v}}_{\mathbf{1}}}=\frac{8}{5}\mathbf{i}+\frac{4}{5}\mathbf{j}$, ${{\mathbf{v}}_{\mathbf{2}}}=\frac{7}{5}\mathbf{i}-\frac{14}{5}\mathbf{j}$.